#7806710

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#43019	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,20)(13,15)(16,18)(17,22)(19,24)(21,23)(25,26,29,33)(27,32,34,30)(28,31,37,40)(35,39,36,38);(1,3,5,8)(2,6,4,7)(9,11,13,18)(10,14,15,16)(12,21,22,19)(17,24,20,23)(25,27,29,34)(26,30,33,32)(28,36,37,35)(31,39,40,38);(10,15)(11,18)(17,20)(19,21)(27,34)(30,32)(35,36)(38,39);(9,11)(10,16)(12,19)(13,18)(14,15)(17,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,19)(13,22)(14,23)(15,20)(16,24)(18,21)(25,28,29,37)(26,31,33,40)(27,35,34,36)(30,38,32,39);(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,22)(21,23)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713906);